New uniqueness results for fractional differential equation with dependence on the ﬁrst order derivative

In this paper, we study the uniqueness of solutions for a fractional diﬀerential equation with dependence on the ﬁrst order derivative. By means of Banach’s contraction mapping principle and a weighted norm in product space, suﬃcient conditions for the uniqueness of solutions are investigated. An example is given to illustrate the main results.

The problem of the existence of solutions for fractional differential equation with various boundary conditions has received an increased attention by using variational methods and critical point theory, the theory of coincidence degree, some well-known fixed point theorems, upper and lower solution method; see the monographs of Kilbas et al. [17], Miller and Ross [21], Podlubny [23], the papers [1-5, 12, 14-16, 18, 19, 24-27, 29-31, 33, 35, 36, 38, 40-42], and the references therein. For example, Bai and Lü [5] considered the special case of BVP (1.1)thatf does not contain first order derivative term u ′ : ). The authors obtained the existence and multiplicity of positive solutions by means of the Krasnosel'skii fixed point theorem and the Leggett-Williams fixed point theorem. In [18], the existence of at least one solution for BVP (1.1) is proved by the Leray-Schauder continuation principle. In [2], the authors investigated the fractional differential equations where 1 < α <2,µ >0arerealnumbers,αµ ≥ 1, f is a Carathéodory function, and f (t, x, y)issingularatx = 0. The authors obtained the existence of positive solutions based on regularization and sequential techniques. Recently, the authors of [9]proveduniqueness results for BVP (1.2) by means of Banach's contraction mapping principle and the theory of linear operator.
At present, many papers are devoted to the uniqueness results for BVP; see [6-11, 13, 20, 22, 28, 34, 37, 39]. Some nonlinear analytical techniques have been used to study the uniqueness of solutions for differential equation and differential systems such as the method of Banach's contraction mapping principle, fixed point theorems for mixed monotone operators, the maximal principle, u 0 -positive operator, and linear operator theory. On the other hand, there are some papers studying fractional differential equations and fractional differential systems in which the fractional orders are involved in the nonlinearity, we refer the reader to [2,18,32]. Motivated by the results above, utilizing Banach's contraction mapping principle, we investigate the uniqueness result for solution of BVP (1.1).
It should noted here that our main result has various new system features. First of all, BVP (1.1) is reformulated as a fixed point problem for system of integral equations. Second, a weighted norm in product space is introduced. Third, the first order derivative is involved in the nonlinear terms.
Throughout the paper, we assume that the following condition holds: (H) f : [0, 1] × R 2 → R is a continuous function and there exist constants A, B >0such that

2P r e l i m i n a r i e s
Define two functions G, G 1 as follows: After routine calculation we get the following four inequalities: and Here, B(α, β) is the beta function defined by the Euler integral: Moreover, u ′ ∈ C(0, 1] ∩ AC loc (0, 1], lim t→0 + t 2-α u ′ (t) exists and satisfies We set E 1 = C[0, 1] with the usual maximum norm denoted by u E 1 = max t∈(0,1] |u(t)|. Consider the Banach space According to Lemma 2.2,BVP(1.1)hasasolutionu = u(t)ifandonlyif(u, v) ∈ E 1 × E 2 solves the following integral equations:  with v = u ′ . Define an operator T by where operators T 1 , T 2 are defined by respectively. For (u, v) ∈ E, by Lemma 2.1 and (H), we have which implies that T 1 is well defined on E and T 1 (u, v) ∈ E 1 .Inthesameway,wecanprove that T 2 is well defined on E and T 2 (u, v) ∈ E 2 for all (u, v) ∈ E. Thus, the existence of a solution of BVP (1.1)isequivalenttotheexistenceofafixedpointofT on E 1 × E 2 . It follows from (2.7)thatT 1 maps all of E 1 × E 2 into the following vector subspace of E 1 : Clearly, E 3 is a Banach space with the norm Hence, in the following we only need to consider the fixed points of T in the Banach space with a constant θ >0.

The above inequality is equivalent to
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